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Growing Waves Due to Transverse 
Velocities 

By J. R. PIERCE and L. R. WALKER 

(Manuscript received March 30, 1955) 

This paper treats propagation of slaw ivavcs in two-dimensional neu- 
tralized electron flow in which all electrons have the same velocity in the 
direction of propagation but in which there are streams of two or more veloci- 
ties normal to the direction of propagation. In a finite beam in which 
electrons are reflected elastically at the boundaries and in which equal dc 
currents are carried by electrons with transverse velocities +Mi and — U\ , 
there is an antisymmctrical growing wave if 

co p 2 ~ (WHO* 

and a symmetrical growing wave if 

Here w p is plasma frequency for the total charge density and W is beam 
width. 

INTRODUCTION 

It is well-known that there can be growing waves in electron flow when 
the flow is composed of several streams of electrons having different 
velocities in the direction of propagation of the waves. 1 " 5 While Birdsall 6 
considers the case of growing waves in electron flow consisting of streams 
which cross one another, the growing waves which he finds apparently 
occur when two streams have different components of velocity in the 
direction of propagation. 

This paper shows that there can be growing waves in electron flow 
consisting of two or more streams with the same component of velocity 
in the direction of wave propagation but with different components of 
velocity transverse to the direction of propagation. Such growing waves 
can exist when the electric field varies in strength across the flow. Such 
waves could result in the amplification of noise fluctuations in electron 
flow. They could also be used to amplify signals. 

109 



110 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

Actual electron flow as it occurs in practical tubes can exhibit trans- 
verse velocities. For instance, in Brillouin flow, 7 « 8 if we consider electron 
motion in a coordinate system rotating with the Larmor frequency we 
see that electrons with transverse velocities are free to cross the beam 
repeatedly, being reflected at the boundaries of the beam. The trans- 
verse velocities may be completely disorganized thermal velocities, or 
they may be larger and better-organized velocities due to aberrations at 
the edges of the cathode or at lenses or apertures. Two-dimensional 
Brillouin flow allows similar transverse motions. 

It would be difficult to treat the case of Brillouin or Brillouin-like flow 
with transverse velocities. Here, simpler cases with transverse velocities 
will be considered. The first case treated is that of infinite ion-neutra- 
lized two-dimensional flow with transverse velocities. The second case 
treated is that of two-dimensional flow in a beam of finite width in which 
the electrons are elastic-ally reflected at the boundaries of the beam. 
Growing waves are found in both cases, and the rate of growth may be 

large. 

In the case of the finite beam both an antisymmetric mode and a 
symmetric mode are possible. Here, it appears, the current density 
required for a growing wave in the symmetric mode is about % times 
as great as the current density required for a growing wave in the anti- 
symmetric mode. Hence, as the current is increased, the first growing 
waves to arise might be antisymmetric modes, which could couple to a 
symmetrical resonator or helix only through a lack of symmetry or 
through high-level effects. 

1 . Infinite two-dimensional flow 

Consider a two-dimensional problem in which the potential varies 
sinusoidally in the y direction, as exp(-j/3z) in the z direction and as exp 
(jtd) with time. Let there be two electron streams, each of a negative 
charge p and each moving with the velocity u in the z direction, but 
with velocities «i and — Ui respectively in the y direction. Let us denote 
ac quantities pertaining to the first stream by subscripts 1 and ac quan- 
tities pertaining to the second stream by subscripts 2. The ac charge 
density will be denoted by p, the ac velocity in the y direction by y, 
and the ac velocity in the z direction by z. We will use linearized or 
small-signal equations of motion. 9 We will denote differentiation with 
respect to y by the operator D. 

The equation of continuity gives 

jcopi = -D(pi«i + poyi) + j/3(pit/o + Poii) (LI) 

jwp 2 = —D(-p 2 ih + Pom) + j/3(p2«o + P0Z2) (1.2) 



% 



GROWING WAVES DUE TO TRANSVERSE VELOCITIES 111 

Let us define 

<h = j'(« - 0J/ O ) + u x D (1.3) 

(h = j(u - /3«o) - w,D (1.4) 

We can then rewrite (1.1) and (1.2) as 

<hp\ = Po(-Dyi + jfci) (1.5) 

d2pa = Poi-Dfa + jiara) (1.6) 

We will assume that we are dealing with slow waves and can use a po- 
tential V to describe the field. We can thus write the linearized equations 
of motion in the form 

diii = -j-PV (1.7) 

m 

d&. = -J-PV (1.8) 

m 

diyi = - DY (1.9) 

m 

chyi = - DV (1.10) 

in 

From (1.5) to (1.10) we obtain 

di'pi = --po(Z) 2 - /3 2 )7 (1.11) 

m 

r / 2 2 P2= -1 P0 {D 2 - p 2 )V (1.12) 

m 

Now, Poisson's equation is 

{If - tf)v = _^_±^. 2 (1.13) 

e 

From (1.11) to (1.13) we obtain 

(D 2 - /r')F = - V 2 ^ (JL + 1) (Z) 2 - ^ 2 )F (1.14) 

, ~ 2 J P ° (1.15) 

oj ;) — 

e 

Here co p is the plasma frequency for the charge of both beams. 



112 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

Either 

(D 2 - /3 2 )7 = 



or else 



1 = 



-up (d 2 + di) 



2 di 2 d 2 2 

We will consider this second case. 

We should note from (1.3) and (1.4) that 

d? = u{D 2 - (w - 0«o) 2 + 2jD(u - /3w )«i 

di = UiD* - (w - Quo? - 2jD(co - /3«o)mi 

d* + f /, 2 = 2[»! 2 Z) 2 - (to - |3wo) 2 ] 

dkW - [wi 2 D 2 + (- - /3«o) 2 ] 2 

Thus, (1.17) becomes 

- <*,%& ~ (» - Z 3 ^) 2 ! 

[mi 2 Z> 2 + (co - 0-m o ) 2 ] 2 

If the quantities involved vaiy sinusoidally with y as cos yy or sin yy, 
then 



(1.16) 

(1.17) 

(1.18) 
(1.19) 

(1.20) 
(1.21) 

(1.22) 



D 2 = -y- 



Our equation becomes 



1 = J^L 
y-Ui 



1 + 



7Wi 



-~i 



1 - 



yui J 



(1.23) 



(1.24) 



What happens if we have many transverse velocities? If we refer back 
to (1.14) we see that we will have an equation of the form 



i-E 



Vl^pn 



2 f din + din 

din' din 



(1.25) 



Here o> pn 2 is a plasma frequency based on the density of electrons having 
transverse velocities ±u n . Equation (1.25) can be written 



y-u n - 



1 + 


(« 


- /3i/o) 2 " 




7 2 Mn 2 _ 


1 - 


(CO 


- 0Mo) 2 ' ' 




7 2 M n 2 



(1-26) 






GROWING WAVES DUE TO TRANSVERSE VELOCITIES 



113 




(O>-/3U 



Fig. 1 

Suppose we plot the left-hand and the right-hand sides of (1.26) versus 
(co — (iiio). The general appearance of the left-hand and right-hand sides 
of (1.2G) is indicated in Fig. 1 for the case of two velocities u n . There 
will always be two unattenuated waves at values of (w — /3w ) 2 > y 2 u e 2 
where u e is the extreme value of ii„; these correspond to intersections 3 
and 3' in Fig. 2. The other waves, two per value of u n , may be unat- 
tenuated or a pair of increasing and decreasing waves, depending on the 
values of the parameters. If 



> 1 



there will be at least one pair of increasing and decreasing waves. 

It is not clear what will happen for a Maxwellian distribution of veloci- 
ties. However, we must remember that various aberrations might give a 
very different, strongly peaked velocity distribution. 

Let us consider the amount of gain in the case of one pair of transverse 
velocities, ±«i . The equation is now 



2 2 

7 "i 



>♦(* 


- 0woY' 

7^1 / _ 


:>-(•- 


- 0ttoY" ' 
yui ) 



(1.27) 



Let 



« + .7^f 

U Wo 



(1.28) 



114 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



0.9 



0.7 



0.5 




0.3 



0.2 



0.1 



Fig. 



This relation defines e. Equation (1.27) becomes 



yui 



1 - € 



(1.29) 
«,* (1 + e 2 ) 2 

In Fig. 2, e is plotted versus the parameter -fu*/wj. We see that as the 
parameter falls below unity, c increases, at first rapidly, and then more 
slowly, reaching a value of ±1 as the parameter goes to zero (as w p 
goes to infinity, for instance) . 

It will be shown in Section 2 of this paper that these results for infinite 
flow are in some degree an approximation to the results for flow in narrow 
beams. It is therefore of interest to see what results they yield if applied 
to a beam of finite width. 

If the beam has a length L, the voltage gain is 

e TC*i/*4£ (1.30) 

The gain G in db is 

(1.31) 



G = 8.7^edb 



wo 



GROWING WAVES DUE TO TRANSVERSE VELOCITIES 115 

Let the width of the beam be W. We let 

y - w e- 32 > 

Thus, for n = 1, there is a half-cycle variation across the beam. From 
(1.31) and (1.32) 

G = 27.3 (-=\ru db (1.33) 

\Mo a / 

Now L/uq is the time it takes the electrons to go from one end of the 
beam to the other, while W/u\ is the time it takes the electrons to cross 
the beam. If the electrons cross the beam N times 

N ~lw wo 

Thus, 

G = 27.3 Nne db (1.35) 

While for a given value of e the gain is higher if we make the phase 
vary many times across the beam, i.e., if we make n large, we should 
note that to get any gain at all we must have 

2 . 2 2 

co p > 7 U\ 
2 ^ /mrwiY (1-36) 

If we increase w p , which is proportional to current density, so that w p 2 
passes through this value, the gain will rise sharply just after « p 2 passes 
through this value and will rise less rapidly thereafter. 

2. A Two-Dimensional Beam of Finite Width. 

Let us assume a beam of finite width in the y-direction; the boundaries 
lying at y = ±?/o . It will be assumed also that electrons incident upon 
these boundaries are elastically reflected, so that electrons of the incident 
stream (1 or 2) are converted into those of the other stream (2 or 1). The 
condition of elastic reflection implies that 

fa = -2/2 (2.1) 

Zi = Z2 at y = ±?/o (2.2) 

and, in addition, that 

Pi = p 2 at y = ±?/o (2.3) 

since there is no change in the number of electrons at the boundary. 



116 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

The equations of motion and of continuity (1.7-1.12) may be satisfied 
by introducing a single quantity, \f/, such that 



V = dx diif 




(2.4) 


iy = —j — 8 d\ d% ty 
m 




(2.5) 


zi = — j — di dvj/ 
m 




(2.6) 


Vi = — di d^Dyf/ 
m 




(2.7) 


yi= — d\ drfty 

m 




(2.8) 


w = -- P0 (D 2 - 8 2 ) cfaV 
m 


(2.9) 


P2 = -1 UD 2 - 8 2 ) 


dxY 


(2.10) 



m 
Then, if we introduce the symbol, ft, for w — 8u 



yi + y2 = 2j - di. d 2 Dty (2.11) 



m 



h- 22 = 2j-did*uiDf (2-12) 

Pl - P2 = 2j- po(D 2 - 8 2 )uSlD+ (2.13) 

m 

It is clear that if 

Zty = 2>V = y = ±yo ( 2 - 14 ) 

the conditions for elastic reflection will be satisfied. The equation satis- 
fied by \f/ may now be found from Poisson's equation, (1-13), and is 

(D 2 - 8 2 ) d? d,V = — (& ~ /3 2 )(^i 2 + d?)+ 
me 

or 

(D 2 - 8 2 )[{u x 2 D 2 + ft 2 ) 2 + co p 2 ( Wl 2 D 2 - O 2 )] = (2.15) 

which is of the sixth degree in D. So far four boundary conditions have 
been imposed. The remaining necessary pair arise from matching the 



GROWING WAVES DUE TO TRANSVERSE VELOCITIES 117 

internal fields to the external ones. For y > y 

V = Voe-^-e- " (2.16) 



and 



Similarly 



5^ + (3V = at y = y 
oy 



^ - 0V - at 7/ - -7/o (2.17) 

The most familiar procedure now would be to look for solutions of 
(2.15) of the form, e". This would give the sextic for c 

(c 2 - rtKWV + a 2 ) 2 + ^(Vc 2 - n 2 )] = o (2.18) 

with the roots c = ±/3, ±ci , ±c 2 , let us say. We could then express \f/ 
as a linear combination of these six solutions and adjust the coefficients 
to satisfy the six boundary equations. In this way a characteristic equa- 
tion for /3 would be obtained. From the symmetry of the problem this 
has the general form F(fi, c,) = F(/3, c 2 ), where a and c 2 are found from 
(2.18). The discussion of the problem in these terms is rather laborious 
and, if we are concerned mainly with examining qualitatively the onset 
of increasing waves, another approach serves better. 

From the symmetry of the equations and of the boundary conditions 
we see that there are solutions for ^ (and consequently for V and p) 
which are even in y and again some which are odd in y. Consider first the 
even solutions. We will assume that there is an even function, ^i(?/), 
periodic in y with period 2// , which coincides with \f/(y) in the open 
interval, — y < y < y n and that \pi(y) has a Fourier cosine series repre- 
sentation: 

00 

Hv) = E c n cos \ n y X„ = — n = 0, 1, 2, • • • (2.19) 
i 2/o 

i/> inside the interval satisfies (2.15), so we assume that rpi(y) obeys 

(D 2 - ?)[(*.?& + lf) J + o>;tu{D 2 - tf)fr 

+ „ (220 > 

- E%- 2m + ly ) 

fn=— oo 

where 5 is the familiar 5-function. Since D\f/ and D'V are required to vanish 
at the ends of the interval and \f/, D 2 \f/ and Z)ty are even it follows that all 



118 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

of these functions are continuous. We assume that i/-i = \f/, Ztyi = D\p, 
Dfyi = Dfy, -Dfyi = Z>V and Dfyi = ^V at tne en ^s of the intervals. 
From (2.20), u^Dfyi -> -V$ as jy -> y . 
Since 

£ 5(7/ - 2m + lft) = s?- + - L (-D B cos X„2/ (2.21) 

-oo *Z/<> I/O 1 

we obtain from (2.20) 
2ynP\ — —' 



W " Wp2) (2.22) 

. o y f-ir cos x - y ^ 

+ V l (/3 2 + An 2 )[(a 2 - Wl 2 Xn 2 ) 2 - «,«(ff + Wl 2 Xn 2 )]/ 

Since 

|Z + pv = (D + Mui*D' + rf)V, 

using (2.4), the condition for matching to the external field, 

dV 

dy 

yields, using D$ - Z)V = and ifc'Dty = - W, the relation 

( Wl 2 D 2 + rf)V» =3^ at y - yo . 

Applying this to (2.22), we then obtain, finally, 

Z/o _ 1 

j§ /3 2 [n 2 - u p 2 ] 

(Q 2 - Ml V)' 



+ 2E 



T - (0 2 + X„ 2 )[(fi 2 - Wl 2 X„ 2 ) 2 - co p 2 (fl 2 + M! 2 X„ 2 )] 



(2.23) 



For the odd solution we use a function, My), equal to \/f(y) in - j/ < 
y < ?/o and representable by a sine series. To ensure the vanishing of Zty 
and Dty at i/ = ±2/0 it is appropriate to use the functions, sin n„y, where 
Mn = ( n + H)t/i/o • The period is now 4y Q and we define My) in y a < 
y < 32/o by the relation ^2(2/) = tf(2j/o - 2/) and in -3y < y < -J/o by 
^2(2/) = iV(— 2?/o — y)- Thus, we write 

CO 

^2(2/) = S rfn Sin Mn2/ Mn = fa + MWft 



^2(2/) will be supposed to satisfy 



GROWING WAVES DUE TO TRANSVERSE VELOCITIES 119 

(D 2 - iflKnW + a 2 ) 2 + UrWl? - oft* 

±» (2.24) 

= 2^ [5(2/ — 4m + I2/0) — 8(y — 4m — lf/o)] 

m= — oo 

The extended definition of fa (outside —y < y < ?y ) is such that we may 
again take xpi = \p,- • ■ ■ , Dfyi = DV at the ends of the interval. w/Dtyi is 
still equal to — }/£ at ?/ = // . Now 

+=0 

H [8(.V - 4m + lyo) - £(y - 4m - lj/o)] 

(2.25) 
= — 2 (-1)" sin n„y 

so from (2.24) we may find 

,,.,,, _ _V (-l) n sin M »/y r99(Vl 

{p l + Mn _ )l("" — ttlT*» r — Wp"("" + Ml Z Mn )J 

Matching to the external field as before gives 

(itiD'- + fl 2 )fy 2 = ss at ?/ =yo 
and applied to (2.26) we have 

oo /o 2 2 2 \ 2 

-V± = V (n - Ml Mn ) , 2 2 _v 

2/3 V (/3 2 + Mn 2 )[(fi 2 " WiVn 2 ) 2 - «,*<(* + MlVn*)] ' 

The equations (2.23) and (2.27) for the even and odd modes may be 
rewritten using the following reduced variables. 

.-ft 

7T 

, COT/o Wo 

K = Z 

mil Wi 

7T 2 Mi 2 

(2.23) becomes 



/r 2 - S 2 £=[ 2 2 + n 2 (n 2 - A- 2 ) 2 - 5 2 (n 2 + /c 2 ) 

and (2.27) transforms to 
2 J [(n + W - A- 2 ] 2 



S s 2 + (n + y 2 y- [(n + ' 2 ) 2 - fc 2 ] 2 - 5 2 [(n + J 2 ) 2 + **] ( 2 . 2 9) 

= — 7T2 



120 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



We shall assume in considering (2.28) and (2.29) that the beam is 
sufficiently wide for the transit of an electron from one side to the other 
to take a few RF cycles. The number of cycles is in fact, <ay /rui , and, 
hence, from the definition of z, we see that for values of k less than 2, 
perhaps, z is certainly positive. 

Let us consider (2.29) first since it proves to be the simpler case. If we 
transfer the term xz to the right hand side, it follows from the observa- 
tion that z is positive (for modest values of k), that it is necessaiy to 
make the sum negative. The sum may be studied qualitatively by sketch- 
ing in the k 2 — 5 2 plane the lines on which the individual terms go to 
infinity, given by 



<5 2 = 



[(n + y 2 ) 2 - iff 



(2.30) 




Fig. 3 



GROWING WAVES DUE TO TRANSVERSE VELOCITIES 



121 



0.6 


























/ 
























0.5 

0.4 
0."\ 

























































0.2 0.4 0.6 0.8 1.0 



1.2 1.4 

a/77 



1.6 1. 



2.0 2.2 2.4 



Fig. 4 

Fig. 3 shows a few such curves (n = 0, 1, 2). To the right of such curves 
the individual term in question is negative, except on the line, k 2 = 
(ft + \$) , where it attains the value of zero. Approaching the curves 
from the right the terms go to — oo . On the left of the curves the func- 
tion is positive and goes to + °° as the curve is approached from the 



... 






















/n = 2 


























































/n=i 




































/x\ 


= 


























y 


















z x 


< 
















>< 




\ 


^ 













3 4 5 

Fig. 5 



122 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 



1956 






left. Clearly in the regions marked + which lie to the left of every curve 
given by (2.30), the sum is positive and we cannot have roots. Let us 
examine the sum in the region to the right of the n = curve and to the 
left of all others. On the line, A; 2 = }>i, the sum is positive, since the first 
term is zero. On any other line, A: 2 = constant, the sum goes from + co 
at the n = 1 curve monotonically to — » at the n = curve, so that 
somewhere it must pass through 0. This enables us to draw the zero- 
sum contours qualitatively in this region and they are indicated in Fig. 3. 
We are now in a position to follow the variation in the sum as k varies 
at fixed 5 2 . It is readily seen that for 5 2 < 0.25, because —irz is negative 
in the region under consideration, there will be four real roots, two for 
positive, two for negative k. For 5 2 slightly greater than 0.25, the sum has 




Fig. 6A 



GROWING WAVES DUE TO TRANSVERSE VELOCITIES 123 

a deep minimum for k = 0, so that there are still four real roots unless z 
is very large. For z fixed, as 8" increases, the depth of the minimum de- 
creases and there will finally occur a 8 2 for which the minimum is so shal- 
low that two of the real roots disappear. Call 2(0) the value of z for k = 0, 
write the sum as S(5 2 , fc 2 ) and suppose that S(5o 2 , 0) = —irz(0), then for 
small k we have 

S(S 2 , AT) = -«(0) + (6 2 - 8 2 ) §, + k 2 % = -«(0) -£ * 

do- dk~ Wo 

as 

as as 

aw afc 2 



'as 




y a? 



— (6- - 5 -) + 



. 2 aP 



The roots become complex when 

8 2 = 5 2 - 



(Ui/UoY 



as as 

as 2 ok- 



Since Wi/wo may be considered small (say 10 per cent) it is sufficient to 
look for the values of 5o . 
When k 2 = we have 



-7TZ = 22 



(n + ht 



2 2 + (n + ^)« (n + l AY ~ & 



* t(^4^ + 



z 2 + 5 2 V \(n + ^) 2 - 5 2 (n 4- H) 2 + * 2 
(5 tan 7r5 + 2 tanh xz) 



z 2 + 5 



Fig. 4 shows the solution of this equation for various 2(0) or coy /iruo • 
Clearly the threshold 5 is rather insensitive to variations in uyo/inio • 

Equation (2.28) may be examined by a similar method, but here some 
complications arise. Fig. 5 shows the infinity curves for n = 0, 1, 2, 3; 
the n = term being of the form k /k — 8 . The lowest critical region 
in 8 2 is the neighborhood of the point k = 8 = l /z, which is the intersec- 
tion of the n = and n = 1 lines. To obtain an idea of the behavior of 



124 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



the left hand side (l.h.s.) of (2.28) in this area we first see how the point 
/,- 2 = 5 2 = }i can be approached so that the l.h.s. remains finite. If we 
put A; 2 = V3 + £ and a 1 = V3 + cc and expand the first two dominant 
terms of (2.28), then adjust c to keep the result finite as e — > we find 

- l 3g2 ~ 5 

C " I 3z 2 + 1 

c varies from - % to ! 4 as z goes from to 00 , changing sign at z~ = %. 
Every curve for which the l.h.s. is constant makes quadratic contact with 



the line 5" — V3 = c(k 



•- _ u 



) a t k = 5 = V3. If we remember that 



the l.h.s. is positive for k s = 0, < 8' < 1 and for V = 1, < 5" < 1, 









n = o/^ 


SHADED AREAS jf 
NEGATIVE ff 

Jsk"' f 1 

S S 1 


n = i\ 



1 
3 



Fig. 6B 



GROWING WAVES DUE TO TRANSVERSE VELOCITIES 125 

since there are no negative terms in the sum for these ranges and again 
that the l.h.s. must change sign between the n = and n = 1 lines for 
any k 2 in the range < k 2 < 1 (since it varies from T <» to ± °°), this 
information may be combined with that about the immediate vicinity 
of S 2 = k 2 = % to enable us to draw a line on which the l.h.s. is zero. 
This is indicated in Figs. 6A and 6B for small z and large z respec- 
tively. It will be seen that the zero curve and, in fact, all curves on which 
the l.h.s. is equal to a negative constant are required to have a vertical 
tangent at some point. This point may be above or below k = V% (de- 
pending upon the sign of c or the size of z) but always at a 8 > V$. For 
8 2 < )i there are no regions where roots can arise as we can readily see 
by considering how the l.h.s. varies with k 2 at fixed 8 2 . For a fixed 8 > \i 
we have, then, either for k 2 > \i or A; 2 < M, according to the size of z, 
a negative minimum which becomes indefinitely deep as 8 — » V§. Thus, 
since the negative terms on the right-hand side are not sensitive to small 
changes in 5 2 , we must expect to find, for a fixed value of the l.h.s., two 
real solutions of (2.28) for some values of 5 2 and no real solutions for some 
larger value of 5 2 , since the negative minimum of the l.h.s. may be made 
as shallow as we like by increasing 8 2 . By continuity then we expect to 
find pairs of complex roots in this region. Rather oddly these roots, which 
will exist certainly for 8 2 sufficiently close to V3 + 0, will disappear if 
S 2 is sufficiently increased. 

REFERENCES 

1 L. S. Nergaard, Analysis of a Simple Model of a Two-Beam Growing-Wave 
Tube, RCA Review, 9, pp. 585-601, Dec, 1948. 

2. J. R. Pierce and W. B. Hebenstreit, A New Type of High-Frequency Amplifier, 

B. S. T. J., 28, pp. 23-51, Jan., 1949. 

3. A. V. Haeff, The Electron-Wave Tube — A Novel Method of Generation and 

Amplification of Microwave Energy, Proc. I.R.E., 37, pp. 4-10, Jan., 1949. 

4. G. G. Macfarlane and H. G. Hay, Wave Propagation in a Slipping Stream of 

Electrons, Proc. Physical Society Sec. B, 63, pp. 409-427, June, 1950. 

5. P. Guenard and H. Huber, Etude Expenmentale de L'Interaction par Ondes 

de Charge d'Espace au Sein d'Un Faisceau Electronique se Deplacant dans 
Des Champs Electrique et Magn^tique Crois^s, Annales de Radioelectricite, 
7, pp. 252-278, Oct., 1952. 

6. C. K. Birdsall, Double Stream Amplification Due to Interaction Between Two 

Oblique Electron Streams, Technical Report No. 24, Electronics Research 
Laboratory, Stanford University. 

7. L. Brillouin, A Theorem of Larmor and Its Importance for Electrons in Mag- 

netic Fields, Phys. Rev., 67, pp. 260-266, 1945. 

8. J. R. Pierce, Theory and Design of Electron Beams, 2nd Ed., Chapter 9, Van 

Nostrand, 1954. 

9. J. R. Pierce, Traveling-Wave Tubes, Van Nostrand, 1950.